Thermodynamic cycles and method for generating electricity

ABSTRACT

An apparatus for performing a thermodynamic cycle comprising: a sample having a ferromagnetic phase transition temperature; means to magnetise the sample above the ferromagnetic phase transition temperature of the sample; and means to cool the sample to a temperature that is below the ferromagnetic phase transition temperature thereof, wherein the demagnetisation of the sample whilst the sample is below the ferromagnetic phase transition temperature thereof causes the generation of an independent magnetic flux. Also disclosed is a method of converting energy, comprising the steps of: providing a sample having a ferromagnetic transition temperature; magnetising the sample while the sample is above the ferromagnetic transition temperature thereof; allowing the sample to demagnetise while the sample is below the ferromagnetic transition temperature thereof, the demagnetisation of the sample causing an independent magnetic flux; and converting at least some of the independent magnetic flux into an electric current. An analogous ferroelectric apparatus and an analogous ferroelectric method are also presented.

This Invention relates to a method and device for converting heat energyinto electricity and refrigeration by the use of materials near theirFerromagnetic or Ferroelectric phase transition point; theelectro-calorific and magneto-calorific effects or Ferrofluids,Ferro-electric Fluids or Liquid Crystals by non-Carnot and Carnotlimited thermodynamic cycles.

Direct conversion of heat into electricity with minimal moving parts isdesirable, and electricity is probably the most versatile manifestationof motive power. There are many schemes and devices for directlyconverting heat into electricity by thermo-couples, Seebeck and Peltierdevices. An example of such a device employing the magneto-calorificeffect is disclosed in U.S. Pat. No. 5,714,829.

U.S. Pat. No. 4,3912,356 discloses a method of controlling thetemperature of an element, and the magnetic field applied thereto, tocause the temperature-magnetism state of the element to traverse a loop.The loop may have a first portion of concurrent substantially isothermalor constant temperature and increasing applied magnetic field, a secondportion of lowering temperature and constant applied magnetic field, athird portion of isothermal and decreasing applied magnetic field, and afourth portion of increasing temperature and constant applied magneticfield.

U.S. Pat. No. 3,073,974 discloses a method of employing a capacitiveelement having differences in specific heat capacity between the chargedand uncharged conditions over a given temperature range, for convertingbetween thermal and electrical energy, comprising the steps of applyingelectrical energy to the element at a given voltage within the giventemperature range to change the specific heat capacity from an unchargedvalue to a charged value, subjecting the element to a source of thermalenergy to vary the temperature of the clement within the temperaturerange while maintaining the element in the charged condition; extractingelectrical energy from the element at a voltage different from thevoltage within the temperature range to change the specific heatcapacity of the element from a charged value to an uncharged value; andcooling the element to effect a change in the thermal energy of theelement, thereby varying the temperature of the element within the giventemperature range while maintaining the element in an unchargedcondition.

All of these schemes operate between two reservoirs-source and sink, andas such they are Carnot cycle limited. Low efficiency is a problem inthese devices when dealing with low enthalpy reservoirs such as oceanheat. It is an object of the present invention to seek to provide athermodynamic cycle and method that alleviates this difficulty.

Accordingly, an aspect of the present invention provides an apparatusfor performing a thermodynamic cycle comprising: a sample having aferrormagnetic phase transition temperature; means to magnetise thesample above the ferromagnetic phase transition temperature of thesample; and means to cool the sample to a temperature that is below theferromagnetic phase transition temperature thereof, wherein thedemagnetisation of the sample whilst the sample is below theferromagnetic phase transition temperature thereof causes the generationof an independent magnetic flux, and wherein the means to magnetise thesample comprises a co-material provided adjacent the sample, theco-material exhibiting a phase transition when a predetermined action isperformed thereon, the apparatus further comprising means to perform theaction on the co-material.

Advantageously, the means to perform the predetermined action comprisesmeans to apply an electrostatic field to the co-material.

Alternatively, the means to perform the predetermined action comprisemeans to tension the co-material.

Preferably, the phase transition exhibited by the co-material is asecond-order phase transition.

Another aspect of the present invention provides an apparatus forperforming a thermodynamic cycle, comprising: a sample that exhibitstemporary magnetic remanence; and means to magnetise the sample within atime period that is less than one tenth of the duration of the cycle,the duration of the cycle being less than one ten thousandth of asecond, wherein demagnetisation of the sample causes the generation ofan independent magnetic flux.

Conveniently, the sample cools during a first portion of thedemagnetisation thereof.

Advantageously, the temperature of the sample increases during a secondportion of the demagnetisation thereof.

Preferably, the apparatus further comprises means to convert at leastsome of the independent magnetic flux into an electric current.

Conveniently, the sample is of a first permeability, and wherein aquantity of a material having a second permeability is provided adjacentthe sample, the first permeability being lower than the secondpermeability.

A further aspect of the present invention provides an apparatus forperforming a thermodynamic cycle comprising: a sample having aferroelectric phase transition temperature; means to polarise theorientation of electric dipoles in the sample at a temperature above theferroelectric phase transition temperature; means to cool the sample toa temperature that is below the ferroelectric phase transitiontemperature thereof, wherein depolarisation of the sample whilst thesample is below the ferroelectric phase transition temperature thereofduring the depolarisation thereof causes the generation of anindependent electric flux.

Advantageously, the electrocalorific effect associated with thepolarisation of the sample heats the sample during polarisation thereof.

Preferably, the sample is at an initial ambient temperature thereofprior to the polarisation thereof.

Conveniently, the means to cool the sample to a temperature that isbelow the ferroelectric phase transition temperature thereof comprise,at least partially, a heat exchange between the sample and ambientsurroundings thereof.

Advantageously, the sample is heated to the ambient temperature duringthe depolarisation thereof.

Preferably, the means to cool the sample to a temperature that is belowthe ferroelectric phase transition temperature thereof comprise, atleast partially, an inverse electrocalorific effect associated with partof the depolarisation of the sample.

Another aspect of the present invention provides an apparatus forperforming a thermodynamic cycle, comprising: a sample that exhibitstemporary electric remanence; and means to polarise the sample within atime period that is less than one tenth of the duration of the cycle,the duration of the cycle being less than one ten thousandth of asecond, wherein the depolarisation of the sample causes the generationof an independent electric flux.

Conveniently, the sample cools during a first portion of thedepolarisation thereof.

Advantageously, the temperature of the sample increases during a secondportion of the depolarisation thereof.

Preferably, the means to polarise the sample comprises a flow ofelectric current.

Alternatively, the means to polarise the sample comprises at least onerotating permanent magnet.

Conveniently, the apparatus further comprises means to convert at leastsome of the independent electric flux into an electric current.

Advantageously, the sample is of a first permittivity and a quantity ofa material having a second permittivity is provided adjacent the sample,the first permittivity being lower than the second permittivity.

A further aspect of the present invention provides a method ofconverting energy, comprising the steps of: providing a sample having aferromagnetic transition temperature; magnetising the sample while thesample is above the ferromagnetic transition temperature thereof;allowing the sample to demagnetise while the sample is below theferromagnetic transition temperature thereof, the demagnetisation of thesample causing an independent magnetic flux; and converting at leastsome of the independent magnetic flux into an electric current.

Advantageously, the method further comprises the step of maintaining anambient temperature in the region of the sample that is higher than theferromagnetic transition temperature thereof.

Preferably, the method further comprises the step of allowing the sampleto cool to the ambient temperature following the magnetisation thereof.

Another aspect of the present invention provides a method of conveningenergy, comprising the steps of: providing a sample that exhibitstemporary magnetic remenance; magnetising the sample, thereby causingthe sample to become magnetised in a time period that is less than onetenth of the duration of the cycle, the duration of the cycle being lessthan one ten thousandth of a second; allowing the sample to demagnetise,the demagnetisation of the sample causing an independent magnetic flux;and converting at least some of the independent magnetic flux into anelectric current.

Conveniently, the step of providing a sample comprises the step ofproviding a ferrofluid.

Advantageously, the step of providing a sample comprises the steps ofproviding a sample having a first permeability, and further providing aquantity of a material having a second permeability adjacent the sample,the first permeability being higher than the second permeability.

Preferably, at least one rotating permanent magnet magnetises thesample.

Advantageously, a carrier operable to carry a flow of electric currenttherethrough magnetises the sample.

Conveniently, the step of magnetising the sample comprises the steps ofproviding a co-material adjacent the sample, which co-material exhibitsa further phase transition when a predetermined action is performedthereon, and providing means to perform the action on the co-material.

Advantageously, the step of providing means to perform the action on theco-material comprises providing means to apply an electrostatic field tothe co-material.

Alternatively, the step of providing means to perform the action on theco-material comprises providing means to apply tension to theco-material.

Advantageously, the magnetising step and the converting step are carriedout by a single means operable to magnetise the sample and to convert atleast some of the independent magnetic flux into an electric current.

A further aspect of the present invention provides a method ofconverting energy, comprising the steps of: providing a sample having aferroelectric transition temperature; polarising the orientation ofelectric dipoles in the sample while the sample is above theferroelectric transition temperature thereof; allowing the sample todepolarise while the sample is below the ferroelectric transitiontemperature thereof, the depolarisation of the sample causing anindependent electric flux; and converting at least some of theindependent electric flux into an electric current.

Preferably, the method further comprises the step of maintaining anambient temperature in the region of the sample that is higher than theferroelectric transition temperature thereof.

Conveniently, the method further comprises the step of allowing thesample to cool to the ambient temperature following the polarisationthereof.

Another aspect of the present invention provides a method of convertingenergy, comprising the steps of: providing a sample that exhibitstemporary electric remenance; polarising the orientation of electricdipoles in the sample in a time period that is less than one tenth ofthe duration of the cycle, the duration of the cycle being less than oneten thousandth of a second; allowing the sample to depolarise, thedepolarisation of the sample causing an independent electric flux; andconverting at least some of the independent electric flux into anelectric current.

Advantageously, the step of providing a sample comprises the step ofproviding a ferro-electric-fluid.

Preferably, the step of providing a sample comprises the steps ofproviding a sample having a first permittivity, and further providing aquantity of a material having a second permittivity adjacent the sample,the first permittivity being lower than the second permittivity.

Conveniently, at least one rotating permanent magnet polarises thesample.

Alternatively, a carrier operable to carry a flow of electric currenttherethrough polarises the sample.

Advantageously, the polarising step and the converting step are carriedout by a single means operable to polarise the sample and to convert atleast some of the independent electric flux into an electric current.

Preferably, the method further comprises the step of providing acirculation system comprising micro-encapsulated material whose meltingpoint is close to an operational temperature range of the sample.

A further aspect of the present invention provides a method ofgenerating electricity according to any of the above methods.

A further aspect of the present invention provides a method of producingan electric current according to any of the above methods.

A further aspect of the present invention provides a method ofrefrigeration.

FIG. 1 shows a graph of magnetisation against temperature for a firstcycle embodying the present invention;

FIG. 2 shows a graph of magnetisation against temperature for a secondcycle embodying the present invention;

FIG. 3 shows the schematic layout of a device embodying the presentinvention;

FIG. 4 shows the schematic layout of another device embodying thepresent invention;

FIG. 5 shows the schematic layout of first field generation equipmentfor use with a device embodying the present invention;

FIG. 6 shows the schematic layout of second field generation equipmentfor use with a device embodying the present invention;

FIG. 7 shows the schematic layout of third field generation equipmentfor use with a device embodying the present invention;

FIG. 8 shows the schematic layout of fourth field generation for usewith a device embodying the present invention;

FIG. 9 shows the schematic layout of fifth field generation equipmentgeneration for use with a device embodying the present invention;

FIG. 10 shows the schematic layout of sixth field generation equipmentfor use with a device embodying the present invention;

FIG. 11 shows the schematic layout of a field cancellation circuit foruse with a device embodying the present invention;

FIG. 12 shows a first arrangement of high permeability material with asample for use with a device embodying the present invention;

FIG. 13 shows a second arrangement of high permeability material with asample for use with a device embodying the present invention;

FIG. 14 shows a third arrangement of high permeability material with asample for use with a device embodying the present invention;

FIG. 15 shows a fourth arrangement of high permeability material with asample for use with a device embodying the present invention;

FIG. 16 shows a fifth arrangement of high permeability material with asample for use with a device embodying the present invention;

FIG. 17 shows a sixth arrangement of high permeability material with asample for use with a device embodying the present invention;

FIG. 18 shows a graph of magnetisation against temperature for a thirdcycle embodying the present invention;

FIG. 19 shows a graph of polarisation against temperature for a fourthcycle embodying the present invention;

FIG. 20 shows a graph of polarisation against temperature for a fourthcycle embodying the present invention;

FIG. 21 shows electrostatic field generation equipment for use with adevice embodying the present invention;

FIG. 22 shows a variation of the electrostatic field generationequipment of FIG. 21;

FIG. 23 shows a graph of shape anisotropy in a prolate ellipsoid ofrotation;

FIG. 24 shows a first graph of current and induction in an exciter coiland a sample embodying the present invention respectively;

FIG. 25 shows a second graph of current and induction in an exciter coiland a sample embodying the present invention respectively;

FIG. 26 shows a plot of the Weiss equation for a sample embodying thepresent invention; and

FIG. 27 shows a plot of increase in temperature against temperature fornickel and iron.

Turning firstly to FIG. 1 a graph of magnetisation versus temperaturefor a first cycle embodying the present invention is shown.

The first cycle employs materials with Magneto-calorific properties,that is materials near their ferromagnetic/paramagnetic phase transitionpoints.

Before the commencement of the first cycle, a sample is initially at anambient temperature, which is above the Ferromagnetic phase transitionpoint (i.e. the Curie temperature) of the first sample.

During a first step of the first cycle, a magnetic field is suddenlyapplied to the first sample. This causes the first sample to becomemagnetised, by aligning the magnetic domains within the first sample. Asthis occurs, the temperature of the first sample rises, due to themagnetocalorific effect associated with the magnetisation, to atemperature that is above the ambient temperature. The magnetocalorificeffect occurs due to a localised drop in entropy in the first sampleresulting from the alignment, or ordering, of the magnetic domainstherein, which is compensated for by an increase in thermal energy, andthis effect is described in more detail in Appendix 2.

Once the rate of change of magnetisation of the first sample falls tozero, the thermal energy developed during the first stage of the firstcycle is radiated away due to the thermal gradient between the firstsample and the surroundings (which are maintained at the ambienttemperature), and this cooling comprises a second step of the firstcycle. As the first sample cools, the magnetisation thereof rises,following the curve H>0 shown in Figure, which shows the relationshipbetween magnetisation and temperature dictated by the Weiss equationμ=B_(s)(h+αμ) in the presence of the magnetic field applied to the firstsample.

Once the sample has cooled to the ambient temperature, the first sampleceases cooling and the system reaches a steady state. At this point, athird step of the first cycle is commenced by the sudden switching offof the applied magnetic field. This causes the magnetisation of thefirst sample to fall. As this occurs, the temperature of the firstsample falls, due to the reverse of the magnetocalorific effectdescribed above in relation to the first step of the first cycle. As themagnetic domains within the first sample begin to move out of alignmentwith one another and become randomly orientated, the order within thefirst sample decreases, giving rise to an increase in entropy within thefirst sample. This increase is compensated for by a drop in the thermalenergy present in the first sample.

The curve H=0 shown on FIG. 1 describes the relationship betweenmagnetisation and temperature dictated by the Weiss equation (givenabove) in the absence of an applied magnetic field. The curve H=0 issimilar in form to the curve H>O described above, however the curve H=0is shifted to the left on the temperature axis of FIG. 1 with respect tothe H>0 curve. As the magnetisation and temperature co-ordinates of thefirst sample reach the curve H=0, the magnetisation and temperatureco-ordinates of the first sample will being to follow the curve H=0, andthis marks the end of the third step in the first cycle. The decrease ofthermal energy during the third step cools the first sample to atemperature that is below the Curie temperature of the first sample. Asthe first sample falls below the Curie temperature thereof, it becomesmagnetised.

A fourth step of the first cycle comprises the demagnetisation andheating of the first sample in accordance with the zero-field Weiss law,and at the end of the fourth step the first sample has the samemagnetisation and temperature coordinates as it did before thecommencement of the first step, i.e. having zero magnetisation and beingat the ambient temperature. The demagnetisation of the first sampleduring the fourth step gives rise to an independent magnetic flux.

The necessary condition for an independent flux is that no externalfield is needed to sustain it. Appendix 1 contains a proof showing thata dependent flux will always lead to zero net electrical work in acycle. Appendix 1 also contains a proof showing that an independent fluxcan appear to be dependent if the slew-rate of the external field islow.

Equation A2.7 in appendix 2 can be remodelled to add the new energy termdeveloped in appendix 3 from equation A3.1. However, specific quantitiesmust be considered so dipole moment is re-cast as the dipole moment perunit volume, I, which is also equal to the solenoidal current densitymultiplied by the cross-sectional area.

Thus, from A3.3$I = {{{BA}\quad ɛ_{0}c^{2}{\quad \quad}{and}\quad {BA}} = \frac{I}{ɛ_{0}c^{2}}}$

The expression BA is substituted into A3.2 for φ and then into A3.1leading to:${\Delta \quad U} = {\frac{n}{ɛ_{0}^{2}c^{4}}\quad \frac{k}{R}I{I}}$

This is then substituted into equation A2.7 leading to:${U} = {{T{S}} + {H{I}} + {\frac{n}{ɛ_{0}^{2}c^{4}}\quad \frac{k}{R}I{I}}}$

Routine manipulation as in appendix 2 in equations A2.7, leads to:$( {\Delta \quad T} )_{s} = {{- \quad \frac{T}{c_{H} + {\frac{n}{ɛ_{0}^{2}c^{4}}\frac{k}{R}{T( \frac{\partial I}{\partial T} )}_{H}^{2}}}}( \frac{\partial I}{\partial T} )_{H}\Delta \quad H}$

Thus, a ‘virtual heat capacity’ exists on step 4 of the first cycle dueto the extra term in the denominator summing with C_(H); the dipolesfind it harder to randomise in the solenoid field around them.Considering the net electrical work in an adiabatic cycle:∮HI = ∫₀^(I_(M))H  I + ∫_(I_(M))⁰H  I = 0

However in the second integral of the thermodynamic cycle, the symmetryis broken: ∫₀^(I_(M))H  I + ∫_(I_(M)^(′))⁰H^(′)  I < 0

For the same ΔT as the adiabatic case, the heat capacity has beenaltered with the ‘virtual heat capacity’. Thus ΔH is not the same, as itmust be higher to maintain the equality since the heat capacity has goneup. This is denoted in the second integral with H′. It is as though twohalf-adiabatic cycles have been pasted together: the first integral isenergy input on the first half of the adiabatic magneto-calorific cycle,the second integral represents the second half of the cycle coming froma much higher magnetisation of a sample with a much higher heatcapacity. This can be reasoned because at the fourth step of the firstcycle, the magneto-calorific effect does not have to ‘know’ or‘remember’ that in the previous step it was allowed to cool isothermallyto ambient. The thermodynamic equations keep no ‘history’ of the changesthey have undergone. What is important are the exact differentials, thethermodynamic co-ordinates of the thermodynamic identity A2.7, at thestart and end of a cycle i.e.

if dU(T,H)=δQ+δW

and ∫dU=0

then if δW<0→δQ>0

The first sample returns to its initial co-ordinates and therefore hasthe same internal energy. However, the first sample did work so musthave absorbed heat.

FIG. 3 shows the essential arrangement of a sample 1, the power coils 2,the field generation equipment 3, rectifier 4, circulation system andheat transfer fluid 5, pump 6 and heat exchanger 7. The sample 1 doesnot have to be localised under the power coils and can travel with theheat exchanger fluid in some embodiments of the invention, however poweris converted under the power coils 2 and field generation equipment 3.

FIG. 4 shows the circulation system replaced by heat pipes 8 in intimatecontact with the sample. The power generation coils 2 can dual as thefield generation equipment 3 (or exciter coils) by appropriateelectrical circuitry. The field generation equipment includes thepossibility of non-electrical field generation by rotating magnets.

The exciter/power coil 2 or field generation equipment 3 surrounding thesample 1 is manipulated in a manner shown by the first cycle to form anindependent flux in the sample 1. The first cycle is traversed manytimes per second. Upon warming this independent flux collapses and poweris delivered to the power coils 2, rectifier 4 and load. Heat energy iscontinually supplied by the circulation system 5, 6, 7 or 8 to thesample. The device thus cools its surrounds.

The electrical loses and gains of the first cycle are modelled here andit is shown that this increases with frequency of operation of thecycle. The resistive losses, loss from not fully recovering the fieldenergy of the exciter stages (steps 1, 2 and 3 of the first cycle) andthe power developed by the cycle are calculated.

For simplicity it is assumed that the current is constant. This will, inany event, tend to overestimate the power loss. The power P_(12R) lostduring the exciter phase is multiplied by ‘D’ (the duty cycle of theon-off period of the square wave). This is frequency independent. Thus:

P ₁ _(²) _(R) =−DI ² R _(coil)

Let B_(E) be the field applied to the core. The energy in the field is½B_(E) ²V where V is the volume of the core. If the back emf isrecovered on field collapse, some of this energy can be recovered. Let ξbe the fraction of this energy wasted. If the cycle is repeated F timesper second:

P _(field)=−ξε₀ C ² B _(E) ² FV

From appendix 3, the power delivered into a resistance R by the cycle isgiven by: $\begin{matrix}{{{Minimun}\quad {Power}} = {\frac{n^{2}k}{R}V^{2}B^{2}F^{2}}} & \text{Eqn.~~A3.6}\end{matrix}$

It will be realised that R is the summation of R_(coil)+R_(load). Byconsidering two resistors in series and using elementary circuit theory,the power dissipated in the resistors as a fraction of the total powerdissipated is R_(x)/(R₁+R₂) where R_(x) is resistor 1 or 2. B_(M) is thefield from the core after the exciter steps. Hence$P_{loss} = {{- {kn}^{2}}V^{2}B_{M}^{2}F^{2}\quad \frac{R_{coil}}{( {R_{coil} + R_{load}} )^{2}}}$$P_{gain} = {{+ {kn}^{2}}V^{2}B_{M}^{2}F^{2}\quad \frac{R_{load}}{( {R_{coil} + R_{load}} )^{2}}}$

The total power is the sum of P_(12R)+P_(field)+P_(loss)+P_(gain). Ifthis is positive there is a net generation of power. It will be notedthat the minimum power generated is squared in F against the field loseswhich are linear. It is simply a matter of engineering to make R_(coil)much less than R_(load).

FIG. 2 shows a graph of magnetisation against temperature for a secondcycle embodying the present invention. Before the commencement of afirst stage of the second cycle a second sample has zero netmagnetisation, and is at an initial ambient temperature. The secondsample is formed from a material having the property of temporarilyretaining magnetic remanence after the application to and subsequentremoval of a magnetic field from the material, for instance aferrofluid.

The first step of the second cycle is commenced by the suddenapplication of a magnetic field to the second sample. This causes themagnetisation of the second sample to increase. In contrast to the firststep of the first cycle, there is very little corresponding rise intemperature as the second sample becomes magnetised, as the initialtemperature is far from the ferromagnetic phase transition point of thesecond sample.

The first step of the second cycle is brought to an end by the abruptremoval of the applied magnetic field from the second sample, this eventbeing a second step of the second cycle.

A third step of the second cycle is commenced immediately after theremoval of the magnetic field from the second sample. During a firstportion of the third step, the temperature of the second sample beginsto fall, as thermal energy within the second sample is expended inworking to re-randomise the magnetic domains in the second sampleagainst the magnetic field arising from the remnant magnetism therein.

The fall in temperature of the second sample during the third stepthereof is initially greatest when the magnetisation is at its highest,at the beginning of the first portion of the third step. This fall intemperature is accompanied by a drop in the magnetisation of the secondsample, as the magnetic domains therein become increasingly non-alignedwith respect to one another.

After a short time, heat from the surrounds warms the sample, and asecond portion of the third step begins. At the same time, the magneticdomains within the second sample become randomly orientated, and thisprovides an independent magnetic flux as the magnetic field arising fromthe alignment of the magnetic domains collapses. At the end of the thirdstep, the magnetisation and temperature co-ordinates of the secondsample are equal to those before the commencement of the first step ofthe second cycle. The rise in temperature, coupled with the fall inmagnetisation, gives rise to the looped shape of the magnetisation curvefor the second sample. For the magnetic flux generated during the secondportion of the third step to be appreciable, the first step preferablyoccurs within the first tenth (or less) of the duration of the secondcycle.

Components employed for field generation during the first step of thesecond cycle and recovery of power from the independent magnetic fluxduring the third step of the second cycle are the same as those employedfor the corresponding tasks in the first cycle, and it will be clear toa skilled person how to put the second cycle into practice.

Listed below are some technical problems that may be encountered inrelation to the first and second cycles, along with their solutions:

Problem Solution Low permeability of sample 1 implies Use of highpermeability materials large exciter field is required hence 13 in closeproximity to a sample 1 much energy expenditure on exciter currentSample 1 has small magnetisation As above Small temperaturedifferentials Increase the surface area of sample Use thin films and/orsmall particles Have circulant 5 or heat pipe 8 in intimate contact withworking substance Boost energy gain per cycle Cycle many thousands oftimes per second Low resistance wiring Use low voltage drop diodes e.g.Schottky or back-diodes High flux linkage e.g. toroid shape Low losshigh permeability materials Minimise eddy current losses by thinsections Large energy expenditure in exciter Recoup exciter field energyby field ‘flyback convertor’ circuit Heat sink current source drivingexciter coil 2 to heat exchanger 7 or Tuned LC circuit Device has largeinductance we Use a step up transformer driving cannot switch rapidly byexciter coil voltage source or allow exciter Lower inductance ofexciter/power field to collapse rapidly coil 2 in power delivery phaseof cycles by use of a lower auto- transformer tap or Shunt a largeinductance into the exciter coil or Switch exciter field with currentsource Respond only to differential changes Use heat pipe 8 to ensurethat in magnetisation in first cycle (FIG. temperature is same form oneend 1) not absolute else the high of device to the other or permeabilitymaterial will saturate. Use field cancellation circuit (Device will havea temperature and/or drop across it) Use high perm. material with‘deadband’ Ferrofluid relaxation time too fast Increase core size orIncrease hydrodynamic radius or Increase viscosity of carrier fluid

The problems will be broken down into sub-systems via examples that canoverall be integrated into one device;

Field generation, Exciter/Power output subsystem

Field Cancellation Circuit

Heat exchange/circulation system

Materials and Field boosting by high permeability materials

Viscosity modification of ferrofluid

A Heat Battery

Non-magnetic field exciter scheme

Ferro-electret and Ferro-electric-fluid schemes

Field Generation, Exciter/Power Output Subsystem

Turning to FIG. 3, the field generation equipment 3 can be electrical ormechanical. Possible mechanical schemes are a rotating magnet in astator arrangement (as shown in FIG. 5). When the magnet 9 is alignedwith the poles 10 maximum flux is transmitted to a sample 1. As shown inappendix 1 this flux must have a relatively sharp rise and fall timecompared to the period. This is achieved by sharp pole design.

FIGS. 6 and 7 show the essential schematics of electrical fieldgeneration by tuned circuit (as shown in FIG. 6) and current source (asshown in FIG. 7). The field generation equipment 3 (as shown in FIG. 3)becomes combined with the power generation coils 2.

The circuit 11 shown in FIG. 6 generates a largely sinusoidal field andon independent magnetic flux collapse power is delivered to the load. Toachieve the sharp rise and fall times of the exciter field highpermeability materials 13 are used in proximity to the sample 1.

Circuit 12 (shown in FIG. 6) is a field cancellation circuit thatensures that the field exposed to the working substance varies betweenzero and a net positive flux. The circuit shown in FIG. 7 achieves thesharp rise and falls times by driving the essentially high inductance ofthe device with a current source 14. To recoup dissipation losses in thecurrent source, it should be connected to the heat exchanger 7.

FIG. 8 shows a brute force approach to establishing the exciter field. Asub-circuit 15 consists of a very large capacitance connected to thelower turns of the power/field generation coil 2,3 in a low resistancecircuit. The inductance is low, allowing the current and hence field tobe rapidly established. A higher tap via a diode returns both the fieldenergy and power from independent flux collapse to the capacitor andload.

FIG. 9 shows a more elegant scheme of generating the exciter field byuse of lower currents. More turns are wound so the inductance increases,but a step up transformer is used to lower the inductance as seen by theprimary by L/n². It is ensured that on field collapse a low timeconstant is in effect by tapping the exciter/power generation coil withonly a few turns via switch A (B open) for the dependent part of thefield and then using switch B (A open) for the independent flux. It ispossible to use a separate coil for this that is essentially acylindrical sheet wrapped around the device several times to get goodflux linkage and induced voltage. The diodes are low dropout types suchas Schottky, back-diode or field-effect-transistors with appropriateswitching to ensure rectifying action.

It is possible to wind a high frequency step up transformer withefficiency greater than 99% with careful design. The more field energyrecouped, the more compact the device can be. This is a balance betweenthe P_(field) term and the minimum power expression in the theorysection here in on the power developed. Typical frequencies of operationare 10-500 KHz. The higher limit is set by switching electronics andtypical volumes for the device are in excess of 10 liters. The largeinductance of the device (in the order of Henries) can be reduced by amulti-stage transformer to be in the order of micro-henries and so allowrapid switching via a voltage source. For instance a three-stage 1:1000transformer made from 1:10 will transform the impedance down by 10⁶.Careful design by use of ferrite core in toroidal form (high fluxlinkage), low flux density in the core, low wire losses andspaced/nested low capacitance windings whereby windings are distancedfrom one another by spacers enables this transformer to be constructed.

Another scheme (shown in FIG. 10) to generate a rapidly changing exciterfield is to switch 19 (switch 18 open) current already established in alarger inductor 17 (19 to the left) into the smaller inductance of thefield coil 2,3 (19 to the right). The large inductance should have goodhigh frequency performance and the method discussed earlier for thetransformer applies. When the independent flux collapses switch 18 isclosed and 19 is to the left, field energy and independent flux work aredelivered to the rest of the circuit.

The switching to low inductance of the exciter/power generation coil 2,3on field collapse to lower the time constant is used in all themanifestations of field generation circuitry if not explicitly shown.

Field Cancellation Circuit

Turning to FIG. 11, this procedure is applicable to the first cycle. Aswill be appreciated, when high permeability materials are use to boostthe field from the working substance, the cycle really responds to thechange in magnetisation in the fourth step on the first cycle, not theabsolute magnetisation. FIG. 1 shows a return to the first step at orabove the Curie temperature of the sample 1. To allow operation belowthe Curie temperature and hence a wider temperature range we must notallow this ‘base remenance’ to saturate the high permeability material.Also, the device will have a temperature differential across it with theworking substance at different magnetisation from end to end. A solutionto this is to lower this temperature differential by use of a heat pipe8 (as shown in FIG. 4) which has extremely high conductivity.

A small weak field counter to the base remenance provides a ‘zero level’for the high permeability material. There are two methods of providingthis field: reuse the field/power coils 2,3 and add a biasing currentcircuit 12 to it; or wind an additional coil.

There are two methods of generating the control signal to the fieldcancellation gear: measure the field with a Hall sensor or detect poweroutput, and boost the counter field when this falls. Both signals arelow pass filtered. FIG. 11 shows the general scheme, the sensors beingput into a tight negative feedback loop with the field cancellationgear. FIG. 11 also shows the possibility of winding a cancellation coil14 that has a progressive increase in the field strength in the flowsense of heat transfer fluid 5 through the device. This allows for atemperature differential across the device where there is a higher basemagnetisation on the flow output side (cold side) compared to the flowinput side (warmer side).

Heat Exchange/circulation System

FIGS. 3 and 4 show the circulation fluid 5 bathing or heat pipes 8 inintimate contact with a sample 1. The fluid 5 is inert and has good heatcapacity and transfer characteristics. An example of such a fluid is lowfraction 200/1cS polydimethylsiloxane silicone fluid produced by DowCorning, should the working substance be reactive to water. If thesample in not localised under the coils it can be suspended in thefluid. The next section covers this (as one approach) and alsoferrofluid is a manifestation of the working substance integral with thefluid 5.

To ensure rapid heat flow one must make the working substance have alarge surface area and this concern is addressed below. The fluid 5 canassist rapid flow of heat to the sample 1 by having local heat sources.One can micro-encapsulate a material at its melting point near to theoperating point of the working substance so that there are localisedheat reservoirs. Fats and medium weight alkanes can be encapsulated bycommon methods.

Another method of permitting high heat flow is to use heat pipes whichhave conductivities over a thousands times greater than copper. A fineheat pipe with high surface area in intimate contact with the workingsubstance can be arranged as part of the high permeability fieldboosting materials and this is covered in the next section. The highpermeability materials can be cast into fine pipes, their internalsbeing the heat pipe and the external surface in contact with the workingsubstance. If the operation temperature of the device is kept lower thanthe ambient temperature then heat exchanger design is less demanding.

Materials and Field Boosting by High Permeability Materials

Suitable materials with high magnetic entropy and suitable temperatureranges for the first cycle are Gadolinium (Curie point 16° C.),Copper-Nickel alloys (20-50° C., 28-34% Cu respectively, typically 1-2%Fe) or materials whose Curie points are higher or lower depending if theapplication is extreme. A skilled person will be aware of many suitablematerials. Such materials are formed to have high anisotropy by aligningthe easy axis of magnetisation (by drawing) or high shape anisotropy.Materials suitable for the second cycle are iron, cobalt or ferritebased ferrofluids. The higher the particle anisotropy the greater theinduction the ferrofluid will sustain. Cobalt and ferrite ferrofluidstherefore give high inductions. Changing the shape of the ferrofluidparticles to increase shape anisotropy also helps.

The high permeability material fulfils two functions, of concentratingthe exciter field and boosting the magnetisation from the workingsubstance (especially the first cycle). In the former case, the moreflux concentrated will means less flux wasted in space not near thesample. Ultimately we want to recover field energy to have a compactdevice and there is no point wasting exciter field around non-samplematerial that will never yield energy. In the latter case the boostingis limited by the anisotropy constant of the sample. The field energy ofthe high permeability material should not exceed the equivalent‘anisotropy field energy’ of the sample or it will break up into domainsto reduce the magneto-static energy. Typical anisotropy fields are ofthe order of tens of kA/m limiting the volume of the high permeabilityto about ten times the volume of the working substance. The anisotropyfield can be enhanced by the ‘shape anisotropy field’ and by making theworking substance long and slender. However the equivalent dipolerepresentation of this long material has the dipoles far apart thusfringing fields into the high permeability material will be low, we mustthen use an even higher permeability material. The relevant designcriterion are shown below:

Typical crystalline (k₁) anisotropies/(kA/m): Gd 50, CuNi alloy 30,Cobalt 60 (ferrofluid)

The shape anisotropy field H_(c) is given by:

H _(c)=(D _(z) −D _(x))M _(s)

where M_(s) is the saturation magnetisation and the demagnetisationfactors are shown in FIG. 23. FIG. 23 shows the shape anisotropy in aprolate ellipsoid of revolution showing variation of the difference inthe two principal demagnetising factors D_(z)−D_(x) as a function of theaxial ratio a/b.

Shape anisotropy is seen to be of little use in the first cycle wheremagneto-calorific materials are used because the magnetisation is so lowanyway, the k₁ anisotropy is primarily relied upon. This is not ofconcern with ferrofluids where large magnetisation is generated.

In the following dipole equations: $\begin{matrix}{B_{r} = {\frac{\mu_{0}}{4\pi}\quad \frac{2p_{m}}{r^{3}}\cos \quad \theta}} \\{B_{0} = {\frac{\mu_{0}}{4\pi}\quad \frac{p_{m}}{r^{3}}\sin \quad \theta}} \\{p_{m} = {ml}}\end{matrix}$

the angular and radial parts of the field are shown. The magnetic dipolemoment is a product of the pole strength and their distance apart.

FIGS. 12-17 showing various arrangements of high permeability material13 with the sample 1. The material 13 is of small cross-section so as tohave low eddy current losses. Suitable materials include iron, nickel,mu-metal, ferrites, vitrovac (Seimens corporation). Variousconfigurations are described below.

FIG. 12 shows an arrangement of small pipes that can dual as heat pipes.One can use electrode-less deposition to deposit iron or nickel ontonylon wire. The wire can be heated to burn off the nylon leaving a highpermeability pipe 13. For materials suitable for use with the firstcycle, a further deposition of the working substance 1 can be done byfurther electrode-less deposition, electroplating, vapour deposition,spraying or electrostatic deposition followed by a binding process.

FIG. 13 shows an arrangement of the high permeability material as aloose wire ‘wool’ matrix or loom that has body and support and can havethe working substance (for the first cycle, ferrofluid will flow intoits structure) deposited by the deposition methods just stated. The heattransfer fluid 5 (as shown in FIG. 3) can pass easily through thestructure. The wire loom form is amenable in preparation by claddingiron or nickel rods with the working substance (if ductile, Cu—Ni, Gd)and then stretching.

FIG. 14 shows an arrangement of the high permeability material as aloose spiral. This configuration is amenable to having the workingsubstance hot pressed onto it.

FIG. 15 shows an alternating sandwich structure of the high permeabilitymaterial with holes 20 therethrough to permit flow of heat transferfluid 5 or heat pipes 8 (as shown in FIGS. 3 and 4).

FIG. 16 shows an arrangement as long, slender entities 21 that can behigh permeability material 13 on working substance 1 or 1 on 13. For thesecond cycle we just have the material 13. These entities will alignthemselves with the flow of heat transfer fluid 5 when dispersed and canthus be made to generate the greatest field in the intended direction.Occasional long bursts of the exciter field also help to align theentities. Methods of manufacture include electrode-less deposition,electroplating, vapour deposition onto wire, cladding and stretchingonto wire, spraying or electrostatic deposition, growth then chemicaltreatment (Ni wire with Cu—Ni alloy around it), and precipitation ofpure phase (Ni in Cu—Ni alloy for instance) by heat or other treatment.

FIG. 17 shows an arrangement of the long slender entities 21 of FIG. 16aligned in a solid block that can be drilled with circulation holes 20as shown in FIG. 15. Around the entities is grown a non-magnetic andnon-electrically conductive material 22 to prevent flux closure and eddycurrent loss. The material 22 can be grown by the methods discussedherein. The plurality of entities can be compacted or sintered or madeto chemically adhere to form the block.

The relaxation time for the independent flux to be destroyed is acharacteristic of the ferrofluid (discussed below) but for the firstcycle materials the dimensions as governed by the Fourier heat diffusionequation. The time it takes the particles to return to the temperatureof their surroundings is modelled by a simple one-dimensional FourierLaw and to obtain the approximate dimensions, the rest is empiricallyderived.

Thus: ${{mc}\quad {\Delta\theta}} = {{kA}\quad \frac{T}{x}t_{r}}$

Where

m is the mass;

C is the specific heat capacity;

Δθ is the temperature difference;

k is the thermal conductivity;

A is the area for heat transfer;

dt/dx is the spatial rate of change of temperature; and

t_(r) is the time taken for the process.

Let the particle be modelled as a long cylinder of radius r, and lengthh. The ratio of volume to area is r/2 for h>>r. Let ρ be the density ofthe material. Thus,$t_{r} = \frac{\rho \quad {rc}\quad \Delta \quad \theta}{2k\quad \frac{T}{x}}$

Substitution of common average figures will give a ‘ballpark’ figure fort_(r). A figure of about 10 μs is arrived at for dimensions of order of10 μm.

Viscosity Modification of Ferrofluid

Ferrofluid is used as a sample in the second cycle. As explained inappendix 1 the induction from the sample must be independent if there isto be a net gain of energy from the device. Electronic switching will beslew-rate limited and with power electronics will have a typical upperlimit of 1 MHz. The remenance relaxation time of the ferrofluid musttherefore be 10 μs or more. Unmodified ferrofluid has a relaxation timeof the order of 100 picoseconds or quicker. This can be achieved byslowing down the response over several orders of magnitude by threemethods: increasing the core size, increasing the hydrodynamic radius bychanging the size of the surfactant polymer or the addition of viscosityenhancing agents. Processing is then performed to remove most of thespread in relaxation time and keep in a relatively narrow window of theintended relaxation time.

A Heat Battery

With reference to FIGS. 3 and 4 the heat exchanger can be interfaced toa system of high heat capacity and temperature to form a heat battery.This system could be a storage tank or a small furnace offering hybridoperation whereby fuel is burned to generate electricity. Thus, an on‘demand unit’ can be made in which the tank ‘trickle charges’ from asource with a low rate of heat transfer to the device which can, forbrief bursts, permit a much greater heat flow.

Non-magnetic Field Exciter Scheme

FIG. 18 shows a third cycle which is a variant on the first cycle, wherean electro-static field is used in conjunction with a co-material (beinga liquid crystal or similar material) that undergoes a second orderphase transition when stressed in some manner. Another example is a longchained polymer material subject to mechanical extension.

The manner of thermal cycling in the third cycle is to apply the stress(field, tension etc.) to the co-material so that it warms by secondorder phase transition during a first step of the third cycle. Theco-material then cools to ambient with the field on, this coolingcomprising a second step of the third cycle. The field is then switchedoff so that the co-material drops below the ambient temperature due tothe inverse of the process by which the warming during the second-orderphase transition, during a third step of the third cycle.

Finally, the co-material return to its original temperature during afourth step of the third cycle. In this manner, the co-material cyclesthe third sample.

During the fourth step of the third cycle, an independent flux isgenerated which is destroyed on the temperature rising to ambient. Allprevious methods of arrangement of materials (high surface area, fieldboosting by high permeability materials, methods of heat transfer andcirculation) apply.

FIG. 21 shows an arrangement whereby electrostatic field generatingequipment (3) (a capacitor with the co-material, working substance andheat exchange fluid 5 or heat pipes 8 between its plates) is cycled. Thelarge capacitor 23 on closing switch 25 (all others open) brings thelesser capacitance of the field generating equipment 3 up to the samevoltage. The exciter field is discharged via switch 26 into capacitornetwork 24. The capacitor network 24 is then discharged via switch 27back to the large capacitor 23 to restart the cycle. By this method thefield energy is recouped. The change in magnetisation of the thirdsample is picked up by the power coils 2 as seen in FIGS. 3 and 4. FIG.22 shows the arrangement and means by which the capacitor network 24 isable to return most of its charge to the large capacitor 23. When thefield generation equipment 3 is discharged into 24, all of the switches27 are closed and switches 26 are switched to the chassis so that thecapacitors 25 are in parallel. Upon discharge back to the largecapacitor 23 switches 27 are opened and switches 26 are switched so thatthe capacitors 25 are in series. Fields in excess of 10KV/m andfrequencies of cycling stated earlier are sufficient to polarise theco-material and ensure operation of the device.

Fourth and fifth cycles, shown in FIGS. 19 and 20, are the electrostaticanalogues of the first and second cycles. It will be readily appreciatedhow the polarisation varies with temperature and applied field byanalogy with magnetisation in the first and second cycles.

During a first step of the fourth cycle, a fourth sample, whichinitially has zero net polarisation and is at an ambient temperature(which is above an ferroelectric transition temperature thereof), ispolarised by the application of an external electric field thereto. Thefourth sample heats up during the first step, due to theelectrocalorific effect (analogous to the magnetocalorific effectdescribed herein in more detail), to a temperature above the ambienttemperature.

Once the rate of change of polarisation of the fourth sample has fallento zero, the fourth sample cools to the ambient temperature due to heatexchange with the ambient surroundings thereof This cooling comprises asecond step of the fourth cycle. The polarisation of the fourth sampleincreases as it cools, following the electric equivalent of the‘field-on’ Weiss law (denoted by E>1) described above in relation to thefirst cycle.

Once the fourth sample has reached the ambient temperature, the electricfield is switched off, and the fourth sample begins to depolarise, thetemperature thereof falling during the depolarisation due to the inverseof the electrocalorific effect. This process comprises a third step ofthe fourth cycle, and the third step ends when the polarisation andtemperature co-ordinates of the fourth sample reach the curve dictatedby the electric equivalent of the ‘zero field’ Weiss law (denoted byE=0). At the end of the third stage, the fourth sample is below theferroelectric transition temperature thereof.

The fourth sample then depolarises completely, heating up to the ambienttemperature as it does so. This complete depolarisation causes anindependent electric flux.

Turning to the fifth cycle, which is analogous to the second cycle, afifth sample, which exhibits temporary electric remenance, is at aninitial temperature prior to the commencement of the fifth cycle. Afirst step of the fifth cycle is commenced by the sudden application ofan external electric field to the fifth sample. Once the fifth sample ispolarised, which occurs within the first tenth of the total duration ofthe fifth cycle, the electric field is switched off, which event is thesecond step of the fifth cycle.

The fifth sample depolarises during third step of the fifth cycle, whichis closely analogous to the third step of the second cycle describedabove. The difference to be borne in mind when considering the fifthsample is that it is electric dipoles within the fifth sample that arealigned and subsequently randomised, rather than magnetic domains.

The changing electric field on the fourth step of the fourth cycle andthe third of the fifth cycle causes a changing magnetic field at rightangles to the electric field. This changing magnetic field is picked upby power generation coils arranged at right angles to the electric fieldto deliver power to a load. A suitable material is Barium titanate(BaTiO₃). All previous methods of arrangement of materials (high surfacearea, methods of heat transfer and circulation) apply. The fourth cycleoccurs with the ambient temperature maintained just above theferro-electric phase transition point of a sample employed in the fourthcycle, and it will be clear to a skilled person that an understanding ofthe operation of the first and second cycles can be applied to thefourth and fifth cycles respectively to develop an understanding thereof

In the present specification “comprises” means “includes or consists of”and “comprising” means “including or consisting of”.

The features disclosed in the foregoing description, or the followingclaims, or the accompanying drawings, expressed in their specific formsor in terms of a means for performing the disclosed function, or amethod or process for attaining the disclosed result, as appropriate,may, separately, or in any combination of such features, be utilised forrealising the invention in diverse forms thereof.

Appendix 1—Proof that a Dependent Flux always leads to Net-ZeroElectrical Work in a Cycle

Consider an inductor as some circuit element. The net energy for a cycleis given by: $\begin{matrix}{{\int_{T}{v\quad { \cdot \quad {t}}}} = {- {\int_{T}{\frac{\varphi}{t}{ \cdot \quad {t}}}}}} & \text{Eqn.~~A1.1}\end{matrix}$

The RHS can be integrated by parts: $\begin{matrix}{{\int_{0}^{T}{{(t)}\frac{{\varphi (t)}}{t}\quad {t}}} = {\lbrack {{{i(t)}{\varphi (t)}} - {\int{{\varphi (t)}\frac{{(t)}}{t}{t}}}} \rbrack_{0}^{T} = {{{i(T)}{\varphi (T)}} - {{i(0)}{\varphi (0)}} - {F( {{\varphi (T)},{i(T)}} )} + {F( {{\varphi (0)},{i(0)}} )}}}} & \text{Eqn.~~A1.2}\end{matrix}$

Since i(0) i(T) and φ(0)=φ(T) the first two terms cancel. Let:

i(t)=g(φ(t))   Eqn. A1.3

i.e. a dependent flux, the second integral of equation A2.2 can beintegrated by parts again after the application of the chain rule:$\begin{matrix}{{\int{{\varphi (t)}\frac{{(t)}}{t}{t}}} = {\int{{\varphi (t)}\quad \frac{{g( {\varphi (t)} )}}{{\varphi (t)}}\quad \frac{{\varphi (t)}}{t}{t}}}} & \text{Eqn.~~A1.4}\end{matrix}$

Thus, $\begin{matrix}{{\int_{o}^{T}{{\varphi (t)}\quad \frac{{g( {\varphi (t)} )}}{{\varphi (t)}}\quad {{\varphi (t)}}}} = {\lbrack {{{\varphi (t)}{g( {\varphi (t)} )}} - {\int{{g( {\varphi (t)} )} \cdot 1 \cdot {{\varphi (t)}}}}} \rbrack_{0}^{T} = {{{G( {\varphi (0)} )} - {G( {\varphi (T)} )}} = 0}}} & \text{Eqn.~~A1.5}\end{matrix}$

The first term on the RHS cancels due to the flux being the same at thestart and end of the cycle. The integrand on the RHS cancels for thesame reason. Q.E.D.

How an Independent Flux Becomes a Dependent Flux if Slew Rate if Low

Consider the exciter circuit shown in FIGS. 3 and 4. Turning to FIG. 24,shown to the same scale on the time axis are the current in the excitercoil (I) and the induction in the sample (B). Infinite slew of thecurrent is shown and consequently the decaying induction to the right ofthe switch off point is independent. If, however, the electroniccomponents are slow, the situation shown in FIG. 25 prevails, in whichpart of the B field is not independent and hence not able to do work inexcess of what is put in.

Appendix 2 Background to Magneto-calorific Effect, Ferromagnetism andParamagnetism

Ferromagnetic materials are characterised by several properties:relative permeability greater than unity, spontaneous magnetisation,hysteresis when induction is plotted against incident field—hencedomains and the so called Curie Point Temperature above which, thematerial becomes merely paramagnetic. It is this phase transition to theparamagnetic state that we exploit to turn heat into electricity. First,however, it will be discussed how the spontaneous magnetisation varieswith temperature. Within a domain of a ferromagnetic material, thesaturation magnetisation is modelled well by the Weiss equation eqn.A2.1. Eqn. A2.2 is the Brillouin function. Eqn. A2.3 is the ratio ofaverage spin to total spin that eqn. A2.1 predicts. H is the appliedfield and α is the so-called ‘Molecular field’ related to the exchangeinteraction between the atoms of the sample. $\begin{matrix}{\mu = {B_{S}( {h + {\alpha \quad \mu}} )}} & \text{Eqn.~~A2.1} \\{B_{S} = {{\frac{{2S} + 1}{S}{\coth ( {\frac{{2S} + 1}{S}x} )}} - \quad {\frac{1}{2S}{\coth ( \frac{x}{2S} )}}}} & \text{Eqn.~~A2.2} \\{\mu = \frac{\langle S_{Z}\rangle}{S}} & \text{Eqn.~~A2.3} \\{h = \frac{g\quad \mu_{B}{SH}}{K_{B}T}} & \text{Eqn.~~A2.4} \\{{\alpha \quad T} = {\frac{2S^{2}}{k_{B}T}{pJ}}} & \text{Eqn.~~A2.5}\end{matrix}$

μ_(B) is the ‘Bohr magnetron’ $\begin{matrix}{\mu_{B} = \frac{{e}\hslash}{2m_{e}c}} & \text{Eqn.~~A2.6}\end{matrix}$

where

g is the ‘Landé’ or ‘spectroscopic splitting’ factor;

e is the quantum of electrical charge;

h is Planck's constant dived by 2π;

m_(e) is the mass of the electron,

c is the speed of light;

k_(B) is Boltzmann's constant;

p is the number of nearest neighbours in the lattice; and

J is the exchange energy integral.

FIG. 26 shows a plot of the first equation (h→0) with μ normalisedagainst the maximum value it obtains at zero Kelvin and Kelvintemperature and T normalised against the Curie-temperature T_(c). Itshows how the spontaneous magnetisation falls rapidly near the Curiepoint

The Magneto-calorific Effect

If a ferromagnetic sample is exposed to a magnetic field adiabaticallyits temperature will rise; when the field is removed, it will fall backto the original temperature. We can model this as follows:

dU=TdS+Hdl  Eqn. A2.7

Where H is the magnetic field and I is the magnetisation. If we take Sand H as independent variables, double differentiation gives:$\begin{matrix}{\frac{\partial^{2}U}{{\partial H}{\partial S}} = {( \frac{\partial T}{\partial H} )_{S} = {{- ( \frac{\partial I}{\partial S} )_{H}} = {{- ( \frac{\partial T}{\partial S} )_{H}}( \frac{\partial I}{\partial T} )_{H}}}}} & \text{Eqn.~~A2.8}\end{matrix}$

By definition, the heat capacity at constant field strength is:$\begin{matrix}{c_{H} = {( \frac{\delta \quad Q}{\delta \quad T} )_{H} = {T( \frac{\partial S}{\partial T} )}_{H}}} & \text{Eqn.~~A2.9}\end{matrix}$

Combining this with the previous equation: $\begin{matrix}{{c_{H} = {{- {T( \frac{\partial H}{\partial T} )}_{S}}( \frac{\partial I}{\partial T} )_{H}}}{or}{( {\Delta \quad T} )_{S} = {{- \quad \frac{T}{c_{H}}}( \frac{\partial I}{\partial T} )_{H}\Delta \quad H}}} & \text{Eqn.~~A2.10}\end{matrix}$

The change of magnetisation is at its greatest near the Curie point asthe two graphics show below. Plotted in FIG. 27 is the increase intemperature vs. temperature of the sample (° C. both) for Nickel andIron. The magnetic field shown (H) is measured in Oersteds.

These temperature changes are small, but it will become apparent thatthe cycle can be repeated many thousands of times per second, and muchheat can be converted to electrical energy.

Appendix 3—The Source of Energy on Field Collapse and the Magnitude ofElectrical Power Generated

Consider a coil and load of resistance. Through the coil is anindependent flux φ that changes and starts a current, i, flowing throughthe load R. This arrangement can be modelled as the flux being a dipolein a magnetic field caused by current, i, which itself is caused by theflux changing. The change in energy of the core, that which generatesthe flux is:

ΔU _(core) =−μ·B  Eqn. A3.1

Hence if the flux changes from being parallel to the axis of thesolenoid to being perpendicular thereto, the core loses energy. Thechanging flux induces a current in the solenoid leading to theexpression for B: $\begin{matrix}{B = {{- \quad \frac{n}{ɛ_{0}c^{2}}}\quad \frac{k}{R}{nL}\quad \frac{\varphi}{t}}} & \text{Eqn.~~A3.2}\end{matrix}$

Where:

n is the number of turns per unit length;

L is the length of the coil; and

k is the coupling coefficient 1≧k≧0.

The core can be modelled as a solenoid too, just overlapping the othersolenoid of cross-sectional area A and volume V. The magnetic field fromthis solenoid must be just equal to B (above) from a First Law argument.

The current density through the solenoid is:

j=Bε ₀ c ²  Eqn. A3.3

Hence, the magnetic moment is: $\begin{matrix}{\mu = {{\int_{0}^{L}{{j \cdot A}\quad {l}}} = {j\quad V}}} & \text{Eqn.~~A3.4}\end{matrix}$

The energy dissipated in the resistor is hence: $\begin{matrix}\begin{matrix}{{\Delta \quad U_{R}} = {B\quad ɛ_{0}c^{2}{V \cdot \frac{n}{ɛ_{0}c^{2}}}\quad \frac{k}{R}{nL}\quad \frac{\varphi}{t}}} \\{= {\frac{n^{2}k}{R}V^{2}B\quad \frac{B}{t}}}\end{matrix} & \text{Eqn.~~A3.5}\end{matrix}$

If the thermodynamic cycle is completed F times per second, the rate ofchange of B must at least be B divided by the period, thus:$\begin{matrix}{{{Minimum}\quad {Power}} = {\frac{n^{2}k}{R}V^{2}B^{2}F^{2}}} & \text{Eqn.~~~A3.6}\end{matrix}$

What is claimed is:
 1. An apparatus for performing a thermodynamiccycle, comprising: a sample that exhibits temporary magnetic remanence;and means to magnetise the sample within a time period that is less thanone tenth of the duration of the cycle, the duration of the cycle beingless than one ten thousandth of a second, wherein demagnetisation of thesample causes the generation of an independent magnetic flux.
 2. Anapparatus according to claim 1, wherein the sample comprises a materialwhich cools during a first portion of the demagnetisation thereof.
 3. Anapparatus according to claim 2, wherein the sample comprises a materialin which the temperature of the sample increases during a second portionof the demagnetisation thereof.
 4. An apparatus according to claim 1,wherein the sample comprises a ferrofluid.
 5. An apparats according toclaim 1, which further comprises a circulation system comprisingmicro-encapsulated material whose melting point is close to anoperational temperature range of the sample.
 6. An apparatus accordingto claim 1, further comprising means to convert at least some of theindependent magnetic flux into an electric current.
 7. An apparatusaccording to claim 1, wherein the sample is of a first permeability, andwherein a quantity of a material having a second permeability isprovided adjacent the sample, the first permeability being lower thanthe second permeability.
 8. An apparatus according to claim 1, whereinthe means to magnetise the sample comprises a flow of electric current.9. An apparatus according to claim 1, wherein the means to magnetise thesample comprises at least one rotating permanent magnet.
 10. A method ofconverting energy, comprising the steps of: providing a sample thatexhibits temporary magnetic remenance; magnetising the sample, therebycausing the sample to become magnetised in a time period that is lessthan one tenth of the duration of the cycle, the duration of cycle beingless than one ten thousandth of a second; allowing the sample todemagnetise, the demagnetisation of the sample causing an independentmagnetic flux; and converting at least some of the independent magneticflux into an electric current.
 11. A method according to claim 10,wherein the step of providing a sample comprises the step of providing aferrofluid.
 12. A method according to claim 10, wherein the step ofproviding a sample comprises the step of providing a sample having afi&t permeability, and further providing a quantity of a material havinga second permeability adjacent the sample, the first permeability beinghigher than the second permeability.
 13. A method according to claim 10,wherein at least one rotating permanent magnet magnetises the sample.14. A method according to claim 10, wherein a carrier operable to carrya flow of electric current therethrough magnetises the sample.
 15. Amethod according to claim 10, wherein the magnetising step and theconverting step are carried out by a single means operable to magnetisethe sample and to convert at least some of the independent magnetic fluxinto an electric current.
 16. A method according to claim 10, furthercomprising the step of providing a circulation system comprisingmicro-encapsulated material whose melting point is close to anoperational temperature range of the sample.
 17. A method of producingan electric current, comprising the steps of providing a sample thatexhibits temporary magnetic remenance; magnetising the samples therebycausing the sample to become magnetised in a time period that is lessthan one tenth of the duration of the cycle, the duration of cycle beingless than one ten thousandth of a second; allowing the sample todemagnetise, the demagnetisation of the sample causing an independentmagnetic flux; and converting at least some of the independent magneticflux into an electric current.
 18. A method of refrigeration, the powertherefor being partially provided by the method of claim
 17. 19. Anapparatus for performing a thermodynamic cycle, comprising: a samplethat exhibits temporary magnetic remanence; and means to magnetise thesample within a time period that is less than one tenth of the durationof the cycle, the duration of the cycle being less than one tenthousandth of a second, wherein demagnetisation of the sample causes thegeneration of an independent magnetic flux.